In this paper we consider a quasilinear parabolic equation in a bounded domain under periodic Dirichlet boundary conditions. Our main goal is to prove the existence of extremal solutions among all solutions lying in a sector formed by appropriately defined upper and lower solutions. The main tools used in the proof of our result are recently obtained abstract results on nonlinear evolution equations, comparison and truncation techniques and suitably constructed special testfunction.
"Existence of extremal periodic solutions for quasilinear parabolic equations." Abstr. Appl. Anal. 2 (3-4) 257 - 270, 1997. https://doi.org/10.1155/S1085337597000389